Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and Artifacts

Although the discrete Fourier transform (DFT) is a tool of enormous value in applied mathematics, those who use the DFT are usually on their own learning about its quirks. Books on signal or image processing will sometimes provide comments or cautions in passing about computational details, but they rarely provide any practical advice. Yet many who learn to use the DFT routinely fall into the same pitfalls, and learn (as I did) by making mistakes, often repeatedly. The current book aims to explain the DFT and the various ways one can get into trouble with it. More importantly, it also suggests how to avoid the pitfalls or recognize them and escape any consequent misinterpretation.

The author addresses potential readers who have a basic knowledge of the DFT but may be short of practical experience with it. An introductory chapter reviews the basics of the continuous and discrete Fourier transforms in one, two and many dimensions. The author’s plan after that is to take up one issue at a time, and provide suggestions, examples, and graphical representations in one and two dimensions. His focus is on the DFT itself and not its implementations in fast computational algorithms since the issues he discusses are common across all of them. The style is informal and emphasizes developing good pictorial and intuitive understanding.

Some of the DFT issues are accounting sorts of problems: managing the order of input and input and output sequences properly, and identifying true units for the input and output array. Others are deeper. Aliasing is a well-known problem, and the author has a number of examples — in one and two dimensions — showing the consequences of under-sampling. Less well known is spectral leakage, where the spectrum is blurred or smeared across nearby frequency bins. Spectral leakage is often unavoidable, but its effects can be reduced via data windowing and careful selection of DFT parameters. However, some ways to improve spectral leakage can make aliasing worse, and vice-versa, and the author discusses how one might handle the tradeoffs.

This is not a book for DFT beginners, but it would be useful resource for practitioners and a valuable addition to libraries. It has a good bibliography and a very nice glossary of signal and image processing terms.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.